System and method for performing a Chien search using multiple Galois field elements

ABSTRACT

A system for performing a Chien search simultaneously tests multiple elements of GF(2 P ) as possible roots of a degree-t error locator polynomial σ(x) using a plurality of simplified multipliers that each simultaneously produce the corresponding terms of σ(x). In one embodiment of the system, t−1 simplified multipliers over GF(2 P ) are used to simultaneously test as possible roots α 2 , (α 2 ) 2 , (α 2 ) 3  . . . (α 2 ) j . Each multiplier includes a plurality of adders that are set up in accordance with precomputed terms that are based on combinations of the weight-one elements of GF(2 P ). A summing circuit adds together the associated terms produced by the multipliers and produces j sums, which are then evaluated to test the j individual elements as possible roots. The coefficients of σ(α 2 ) j  are then fed back to the multipliers, and the multipliers test, during a next clock cycle, the elements α 2 *(α 2 ) j , (α 2 ) 2 *(α 2 ) j  . . . , (α 2 ) 2j  and so forth. Similar multipliers also test the odd powers of α as roots of σ′(x)=σ(αx). If P=mn the system may be implemented using a plurality of GF(2 m ) multipliers. The field GF(2 m ) is a subfield of GF(2 P ), and the elements of GF(2 P ) can each be represented by a combination of n elements of GF(2 m ). The error locator polynomial σ(x) can thus be represented by a combination of n expressions σ 0 (x), σ 2 (x) . . . σ n−1 (x), each with coefficients that are elements of GF(2 m ). Each of the n expressions has 2 m −1 coefficients for the terms x 0 , x 1 , x 2  . . . x 2m−1 . Thus, n(2 m −2) constant GF(2 m ) multipliers are used to test each element of GF(2 P ) as a possible root. The number of GF(2 m ) multipliers in the system is independent of the degree of the error locator polynomial, and each multiplier operates over a subfield of GF(2 P ). Accordingly, the system can simultaneously tests j elements using j sets of n(2 m −2) constant multipliers over GF(2 m ).

BACKGROUND OF THE INVENTION

1. Field of the Invention

This invention relates generally to data processing systems and, more particularly, to a system for locating and correcting errors in data using an error correction code.

2. Background Information

Data stored on magnetic media, such as magnetic disks, are typically stored in encoded form, so that errors in the stored data can possibly be corrected. The errors may occur, for example, because of inter-symbol interference, a defect in the disk, or noise. As the density of the data stored on the disk increases, more errors are likely, and the system is required to correct greater numbers of errors. The speed with which the system corrects the errors is important to the overall speed with which the system processes the data.

Prior to recording, multiple-bit data symbols are encoded using an error correction code (ECC). When the data symbols are retrieved from the disk and demodulated, the ECC is employed to, as the name implies, correct the erroneous data.

Specifically, before a string of k data symbols is written to a disk, it is mathematically encoded using an (n, k) ECC to form n-k ECC symbols. The ECC symbols are then appended to the data string to form an n-symbol error correction code word, which is then written to, or stored, on the disk. When the data are read from the disk, the code words containing the data symbols and ECC symbols are retrieved and mathematically decoded. During decoding, errors in the data are detected and, if possible, corrected through manipulation of the ECC symbols [for a detailed description of decoding see, Peterson and Weldon, Error Correction Codes, 2nd Ed. MIT Press, 1972].

To correct multiple errors in strings of data symbols, the system typically uses an ECC that efficiently and effectively utilizes the various mathematical properties of sets of symbols known as Galois fields. Galois fields are represented “GF (S^(P))”, where “S” is a prime number and “P” can be thought of as the number of digits, base “S”, in each element or symbol in the field. S usually has the value 2 in digital computer and disk drive applications and, therefore, P is the number of bits in each symbol. The ECC's commonly used with the Galois Fields are Reed Solomon codes or BCH codes.

There are essentially four major steps in decoding a corrupted code word of a Reed-Solomon code or a BCH code. The system first determines error syndromes that are based on the results of a manipulation of the ECC symbols. Next, using the error syndromes, the system determines an error locator polynomial, which is a polynomial that has the same degree as the number of errors. The system then finds the roots of the error locator polynomial and from each root determines the location of an associated error in the code word. Finally, the system finds error values for the error locations. In binary codes, there is only one possible error value for an error location, and thus, the step of determine g error values is trivial.

The steps of determining the syndromes and finding the error locations are the most time consuming in the error correction process. The invention described herein reduces the time it takes the error correction system to find the error locations, which involves finding the roots of degree-t error locator polynomials

In many systems the roots of the error locator polynomials are determined by trial and error. The trial and error method is performed by substituting into the polynomial every possible value, i.e., every element of the applicable GF(2^(P)) that is associated with a code word location, and for each value evaluating the polynomial. If the polynomial equals zero for a given value, the value is a root. The system continues the trial and error process by substituting a next possible value into the polynomial and determining if that value is a root; and so forth, until either all possible values have been tried or all t roots are determined. This trial and error process, which in an optimized form is commonly known as a Chien search, is time consuming. The time taken to perform the Chien search slows the error correction operations, and thus, the data processing operations of the system.

One way to speed up the Chien search is to test multiple elements simultaneously. However, as discussed with reference to FIG. 1 below, this requires including in the conventional system t−1 additional GF(2^(P)) multipliers for each additional element tested. To test “j” elements simultaneously, the system requires j(t−1) GF(2^(p)) multipliers, and the system thus readily becomes complex.

SUMMARY OF THE INVENTION

A system for performing a Chien search simultaneously tests multiple elements of GF(2^(P)) as possible roots of a degree-t error locator polynomial σ(x) using in one embodiment t−1 simplified multipliers that each simultaneously produce the corresponding terms of σ(x) to test as possible roots α², (α²)², (α²)³ . . . α^(2j). Each multiplier includes a plurality of adders that are set up in accordance with precomputed terms that are based on combinations of the weight-one elements of GF(2^(P)). A summing circuit adds together the associated terms produced by the multipliers and produces j multiple sums, which are then evaluated to test the j individual elements as possible roots. The coefficients of σ(α²)^(j) are then fed back to the multipliers, and the multipliers test, during a next clock cycle, the elements α²*(α²)^(j), (α²)²*(α²)^(j) . . . , (α²)^(2j) and so forth. The same operations are also performed for σ′(x)=σ(αx) to test as possible roots α³, α⁵, α⁷ . . . α^(2j+1).

If P=mn, that is, if P is not prime, the system may be implemented using a plurality of GF(2^(m)) multipliers. The field GF(2^(m)) is a subfield of GF(2^(P)), and the elements of GF(2^(P)) can each be represented by a combination of n elements of GF(2^(m)). The error locator polynomial σ(x) can thus be represented by a combination of n expressions σ₀(x), σ₂(x) . . . σ_(n−1)(x), each with coefficients that are elements of GF(2^(m)). Each of the n expressions has 2^(m)−1 coefficients for the terms x⁰, x¹, x² . . . x^(2m−1). Thus, n(2^(m)−2) constant GF(2^(m)) multipliers are used to test each element of GF(2^(P)) as a possible root, as discussed in more detail below.

The number of GF(2^(m)) multipliers in the system is independent of the degree of the error locator polynomial, and each multiplier operates over a subfield of GF(2^(P)). Accordingly, the system can simultaneously tests j elements using j sets of n(2^(m)−2) multipliers over GF(2^(m)). This is in contrast to conventional Chien search systems that require j(t−1) multipliers over GF(2^(P)) to simultaneously test j elements.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention description below refers to the accompanying drawings, of which:

FIG. 1 is a functional block diagram of a conventional Chien search system;

FIG. 2 is a functional block diagram of a constant multiplier over GF(2^(P));

FIG. 3 is a functional block diagram of a combined GF(2^(P)) for simultaneously multiplying multiple coefficients by a constant multiplier;

FIG. 4 is a functional block diagram of a Chien Search circuit that uses the multipliers of FIG. 3;

FIG. 5 is a functional block diagram of a combined multiplier that uses elements is of GF(2^(m));

FIG. 6 is a functional block diagram of a Chien Search circuit that uses the multipliers of FIG. 5;

FIG. 7 is a functional block diagram of a Chien Search circuit that uses fewer GF(2^(m)) multipliers;

FIG. 8 is a functional block diagram of the Chien Search circuit of FIG. 7 for use with multiple elements of GF(2^(nm)).

DETAILED DESCRIPTION OF AN ILLUSTRATIVE EMBODIMENT

A. A Conventional System

To determine error locations in a code word over GF(2^(P)), a conventional Chien search is performed by individually testing the elements of the Galois Field GF(2^(P)) as roots of an associated error locator polynomial. For an error locator polynomial

σ(x)=σ_(t) x ^(t)+σ_(t−1) x ^(t−1) . . . +σ₁ x+σ ₀

the conventional Chien search system is depicted in FIG. 1. During each clock cycle the registers 10 ₁ . . . 10 _(t) are updated once by multiplying their contents by the corresponding powers of α in GF(2^(P)) multipliers 12 ₁, 12 ₂, . . . 12 _(t). A summing circuit 14 adds the contents of the registers 10 ₀, 10 ₁ . . . 10 _(t), and the system determines, in clock cycle “i,” that the element α^(i) of GF(2^(P)) is a root if the associated sum produced by the summing circuit 14 is zero.

More specifically the registers 10 ₀ . . . 10 _(t) are initially loaded with the corresponding coefficients σ₀, . . . σ_(t) and in a first clock cycle, at time k=0, the field element x=α⁰=1 is tested by calculating in the summing circuit 14 the sum σ_(t)+σ_(t−1)+ . . . +σ₁+σ₀. In the next cycle, at time k=1, the contents of the registers are updated by multiplying them by the associated powers of α in the multipliers 12 ₁, 12 ₂ . . . 12 _(t), and a next sum is calculated by the summing circuit 14 to test the element x=α¹. In the next clock cycle, the element x=α² is tested by again multiplying the contents of the registers by the associated powers of α and adding the results. In the i^(th) clock cycle, the circuit tests the element α^(i) as a root, and so forth. The Chien search system of FIG. 1 thus takes a maximum of k=2^(P)−1 clock cycles to find the t non-zero roots.

One way to speed up the Chien search is to simultaneously test multiple field elements as possible roots. To do this using the system of FIG. 1, however, a set of t−1 GF(2^(P)) multipliers must be included in the system for each additional element tested. Accordingly, 2(t−1) GF(2^(P)) multipliers must be included in the systems to test two elements simultaneously, 3(t−1) GF(2^(P)) multipliers to test three elements, and so forth. To test even two roots simultaneously, the system is thus relatively complex. Discussed below is a system that uses fewer and/or simpler multipliers to simultaneously test multiple field elements as possible roots of the error locator polynomial.

B. A Chien Search System With Simplified Multipliers

An element of a Galois Field GF(2^(p)) can be represented either as a degree-(p−1) polynomial

b _(p−1) x ^(p−1) +b _(p−2) x ^(p−2) + . . . +b ₁ x+b ₀

or as a p component binary vector (b_(p−1), b_(p−2), . . . b₁, b₀) that contains the coefficients of the degree-(p−1) polynomial.

For a Galois Field GF(2⁶) which is generated as successive powers of x mod x⁶+x+1, the field elements in vector form are:

α⁰=1=(000001)

α¹ =x=(000010)

α² =x ²=(000100)

α³ =x ³=(001000)

α⁴ =x ⁴=(010000)

α⁵ =x ⁵=(100000)

α⁶ =x+1=(000011)=(000010)+(000001)

α⁷ =x ² +x=(000110)=(000100)+(000010) . . .

α⁶² =x ⁵+1=(100001)=(100000)+(000001)

α⁶³=α⁰=1=(000001)

Any element c=(c₅c₄c₃c₂c₁c₀) of GF(2⁶) can be represented by a sum of the weight-one elements of GF(2⁶):

c=(c ₅00000)+(0c ₄0000)+(00c ₃000)+(000c ₂00)+(0000c ₁0)+(00000c ₀).

Multiplication operations involving two Galois Field elements α^(i)=(a+b) and α^(j) can be performed as:

α¹*α^(j)=(a+b)*α^(j) =a*α ^(j) +b*α ^(j)

where + and * represent Galois Field addition and multiplication operations, respectively.

Accordingly, the multiplication of c*α^(j) can then be performed as: (c₅00000)*α^(j)+(0c₄0000)*α^(j)+(00c₃000)*α^(j)+(000c₂00)*α^(j)+(00000c₁0)*α^(j)+(00000c₁)*α^(j).

For multiplication of the element c by a constant, such as α^(j)=α², the terms of the product, which are elements of GF(2⁶), can be precomputed as:

c ₅*(100000)*α² +c ₄*(010000)*α² +c ₃*(100000)*α² +c ₂*(010000)*α² +c ₁*(100000)*α² +c ₀*(01000)*α²

multiplying the terms out:

c ₅*[α⁵*α² ]=c ₅*(α⁷)=c ₅*(000110)=c ₅*(000100)+c ₅*(000010)

c ₄*[α⁴*α² ]=c ₄*(α⁶)=c ₄*(000110)=c ₄*(000010)+c₄*(000001)

c ₃*[α³*α² ]=c ₃*(α⁵)=c ₃(100000)

c ₂*[α²*α² ]=c ₂*(α⁴)=c ₂(010000)

c ₁*[α¹*α² ]=c ₁*(α³)=c ₁(001000)

c ₀*[α⁰*α² ]=c ₀*(α^(L))=c ₀(000100)

and combining like terms the product becomes:

c*α ² =c ₃α⁵ +c ₂α⁴ +c ₁α³+(c ₀ +c ₅)α²+(c ₄ +c ₅)α¹ +c ₄

or in matrix form ${\left( {c_{5}c_{4}c_{3}c_{2}c_{1}c_{0}} \right)\quad\begin{bmatrix} 000110 \\ 000011 \\ 100000 \\ 010000 \\ 001000 \\ 000100 \end{bmatrix}} = {{c*A} = \left( {e_{5}e_{4}e_{3}e_{2}e_{1}e_{0}} \right)}$

A Galois Field multiplier that multiplies a field element by the constant α² over GF(2⁶) can thus be implemented by the circuit shown in FIG. 2. As shown in the drawing, the coefficients of c are supplied through registers 18 to Galois Field adders or XOR gates 20. The adders are arranged to add the coefficients together in accordance with the matrix A. The updated coefficients e₅ . . . e₀ are then fed back to the registers 18, and in a next clock cycle the multiplier produces e*α²=c*α⁴.

To calculate (c*α²)*α² or c*α⁴, the terms c*α² are each multiplied by α²:

[c ₃α⁵]*α² +[c ₂α⁴]*α² +[c ₁α³]*α²+[(c ₀ +c ₅)*α²]*α²+[(c ₄ +c ₅)*α¹]*α² +[c ₄]*α²

Multiplying out the terms:

c ₃[α⁷ ]=c ₃(000110)=c ₃*(000100)+c ₃*(000010)

c ₂[α⁶ ]=c ₂(000011)=c ₂*(000010)+c ₂*(000001)

c ₁[α⁵ ]=c ₁(100000)

(c ₀ +c ₅)[α⁴ ]=c ₀(010000)+c ₅(010000)

(c ₄ +c ₅)[α³ ]=c ₄(001000)+c ₅(001000)

c ₄[α² ]=c ₄(000100)

and combing like terms the product becomes:

c*α ⁴ =c ₁α⁵+(c ₀ +c ₅)α⁴+(c ₄ +c ₅)α³+(c ₃ +c ₄)α²+(c ₃ +c ₂)α¹ +c ₂

or in matrix form ${\left( {c_{5}c_{4}c_{3}c_{2}c_{1}c_{0}} \right)\quad\begin{bmatrix} 011000 \\ 001100 \\ 000110 \\ 000011 \\ 100000 \\ 010000 \end{bmatrix}} = {{c*B} = \left( {f_{5}f_{4}f_{3}f_{2}f_{1}f_{0}} \right)}$

Similarly, the product c*α⁴*α² or c*α⁶ can be calculated by multiplying the terms of the previous product by α²:

[c ₁α⁵]*α²+[(c ₀ +c ₅)α⁴]*α²+[(c ₄ +c ₅)α³]*α²+[(c ₃ +c ₄)α²]*α²+[(c ₃ +c ₂)α¹]*α² +[c ₂]α²

and combining like terms:

(c ₄ +c ₅)α⁵+(c ₃ +c ₄)α⁴+(c ₂ +c ₃)α³+(c ₁ +c ₂)α²+(c ₀ +c ₁ +c ₅)α¹+(c ₀ +c ₅)α⁰

which in matrix form is: ${\left( {c_{5}c_{4}c_{3}c_{2}c_{1}c_{0}} \right)\quad\begin{bmatrix} 100011 \\ 110000 \\ 011000 \\ 001100 \\ 000110 \\ 000011 \end{bmatrix}} = {{c*C} = \left( {g_{5}g_{4}g_{3}g_{2}g_{1}g_{0}} \right)}$

The three multiplication operations can then be performed simultaneously by the single matrix multiplication ${{\left( {c_{5}c_{4}c_{3}c_{2}c_{1}c_{0}} \right)\quad\begin{bmatrix} 100011 & \cdots & 011000 & \cdots & 000110 \\ 110000 & \cdots & 001100 & \cdots & 000011 \\ 011000 & \cdots & 000110 & \cdots & 000011 \\ 001100 & \cdots & 000011 & \cdots & 010000 \\ 000110 & \cdots & 100000 & \cdots & 001000 \\ 000011 & \cdots & 010000 & \cdots & 000100 \end{bmatrix}} = \left( {g_{5}g_{4}g_{3}g_{2}g_{1}g_{0}} \right)},\quad \left( {f_{5}f_{4}f_{3}f_{2}f_{1}f_{0}} \right),\left( {e_{5}e_{4}e_{3}e_{2}e_{1}e_{0}} \right)$

which can be implemented in a single combination multiplier circuit as depicted in FIG. 3. The coefficients of c are supplied through registers 28 to adders or XOR gates 30 that are arranged in accordance with the matrix. To perform further multiplication operations, the coefficients g₅ . . . g₀ of the product c*α⁶ are fed back to the registers 28, and in a next clock cycle the adders 30 produce the three products c*α⁸, c*α¹⁰ and c*α¹², and so forth.

The combination multiplier circuit of FIG. 3 can be simplified by eliminating certain adders 30 that correspond to duplicate matrix columns. For example, the columns that produce the coefficients g₅, f₃ and e₁ are identical and the three coefficients can thus be produced by a single adder, or XOR gate. Further simplification may also be realized in certain multipliers by producing coefficients as sums of other coefficients.

As depicted in FIG. 4, a Chien search circuit that simultaneously tests three roots, for example, α², α⁴, and α⁶ may be implemented using t−1 multipliers 40 that are each the simplified Galois Field multipliers discussed above with reference to FIG. 3. Accordingly, each multiplier 40 includes a minimal set of adders 30 (FIG. 3) that are arranged in accordance with the precomputed terms σ′₀, σ′₁ . . . σ′_(t) of the products σ(x) for x=α², α⁴ and α⁶, where σ′₀ represents the precomputed term of x⁰ and so forth. A summer circuit 42 evaluates the three elements as possible roots by adding together the corresponding terms produced by the multipliers 40 to produce three associated sums. If the sum σ(α²)=0, α² is a root. Similarly, if the sum σ(α⁴)=0, α⁴ is a root, and so forth. The coefficients g₅ . . . g₀ of σ(α⁶) are then fed back to the registers 38, and multipliers in a next clock cycle test the next set of three roots α⁸, α¹⁰ and α¹² and so forth.

A similar circuit is used to test the odd powers of α as roots of σ′(x)=σ(αx).

C. A Chien Search System Utilizing Subfields

Even further simplification of the Chien search system of FIG. 4 can be realized if the Galois Field is of the form GF(2^(mn)), and m and n are both small values. The Galois Field GF(2⁶), for example, can be expressed as GF((2²)³) with m=2 and n=3 or GF((2³)²) with m=3 and n=2, where GF(2²) and GF(2³) are subfields of GF(2⁶).

The field GF(2⁶) is isomorphic to a field GF((2³)²) generated by the irreducible polynomial x²+x+1. The subfield GF(2³) has a primitive element β, and is generated as successive powers of β mod β³+β+1. The elements of GF(2³) are:

β⁰=(001)

β¹=(010)

β²=(100)

β³=(011)

β⁴=(110)

β⁵=(111)

β⁶=(101)

and all the elements of GF(2⁶) can be expressed as integer powers of γ, where γ^(k)=β^(j)x+β^(j). For a properly selected γ, there is a one-to-one correspondence between γ^(k) and α^(j), where α is the primitive element of GF(2⁶).

One choice for γ is γ=βx+1. The successive powers of γ, which can be represented as vectors of the form (β^(i), β^(j)), are:

0=(000, 000)=(0,0)

1=γ⁰=(000, 001)=(0, β⁰)

γ=βx+1(010, 001)=(β¹, β⁰)

γ²=(βx+1)(βx+1)=(100, 101)=(β², β⁶)

γ³=(β² x+β ⁶)(βx+1)=(110, 110)=(β⁴, β⁴) . . .

γ⁹=(000, 111)=(0, β⁵) . . .

γ¹⁸=(000, 011)=(0, β³) . . .

γ⁶²=(β³ x+β ⁵)=(011, 111)=(β³, β⁵)

Multiplication over GF(2⁶), for elements γ^(j) of the form (0, β⁵), can performed as the sum of two multiplication operations over GF(2^(m)). Specifically, the multiplication of γ^(k)=(β^(a), β^(b))=β^(a)x+β^(b) and γ⁹=(0, β⁵)=0x+β⁵, may be performed as:

γ^(k)*γ⁹=(β^(a) x+β ^(b))*β⁵=[(β⁵*β^(a))x]+β⁵*β^(b)=(β^(a)*β⁵, β^(b)*β⁵).

Letting β^(a)=d₁ and β^(b)=d₀, the terms d₁ and d₀, which are elements of GF(2³), can be represented as corresponding elements of GF(2⁶) by (d₁, 0) and (0, d₀). The multiplication operation of γ^(k) by the constant γ is then:

γ^(k)*γ=(d ₁, 0)*γ+(0, d ₀)*γ=d₁*(1,0)*γ+d ₀*(0,1)*γ

The element (1,0) in vector form is (001, 000)=γ⁴² and the element (0,1) is (000, 001)=γ⁰. The product thus becomes

(d ₁*γ⁴²*γ)+(d ₀*γ⁰*γ)=d ₁γ⁴³ +d ₀γ=(h ₁ , h ₀)

or in matrix form ${\left( {d_{1},d_{0}} \right)\quad\begin{bmatrix} {\beta^{3} - \beta} \\ {\beta \quad \ldots \quad 1} \end{bmatrix}} = \left( {h_{1},h_{0}} \right)$

with γ⁴³=β³x+β and γ=βx+1.

The product γ^(k)*γ² is (h ₁,h₀)*γ=d₁*γ⁴⁴+d₀*γ²=(m₁, m₀), and the product γ^(k)*γ³ is (m₁, m₀)*γ=d₁*γ⁴⁵+d₀*γ³=(n₁, n₀).

The three multiplication operations can then be performed simultaneously by a single matrix multiplication operation as $d_{1},{{d_{0}\begin{bmatrix} {\beta^{3}\quad \beta} & {1\quad \beta^{2}} & {0\quad \beta^{4}} \\ {\beta \quad 1} & {\beta^{2}\quad \beta^{6}} & {\beta^{4}\quad \beta^{4}} \end{bmatrix}} = \begin{pmatrix} {h_{1},h_{0}} & {m_{1},m_{0}} & {n_{1},n_{0}} \end{pmatrix}}$

A combined multiplier circuit that simultaneously produces the three products γ^(k)*γ, γ^(k)*γ² and γ^(k)*γ³ can be implemented as depicted in FIG. 5, in which the XOR gates of FIG. 3 are replaced with constant GF(2³) multipliers 50. The coefficients n₁, n₀ are fed back to registers 48, and the multiplier circuit in a next cycle produces γ^(k)*γ⁴, γ^(k)*γ⁵, and γ^(k)*γ⁶, and so forth.

The multiplier circuit of FIG. 5 can be simplified by eliminating duplicate multipliers 50. For example, one of the multipliers 50 associated with the n₀ term can be eliminated by using the product d₀β⁴ produced for n₁ also in the sum associated with n₀.

A Chien search system for simultaneously testing three elements of GF(2^(P)) as roots of the error locator polynomial σ(x) over GF(2^(mn)) using constant GF(2^(m)) multipliers 60 is depicted in FIG. 6. The multipliers 60 are each the simplified multipliers discussed above with reference to FIG. 5.

The values σ′^(t), σ′_(t−1) . . . σ′₀ in the form β^(a), β^(b) are supplied to multipliers 60 and at time k=1, the system simultaneously tests γ, γ², γ³ as possible roots. By feeding back to the registers 58 the associated updated coefficients n₁, n₀ produced by the respective multipliers, which are represented in the drawing as n_(i), the system in a next iteration at time k=k+1, tests γ*γ^(3(k−1)), γ²*γ^(3(k−1)) and γ³*γ^(3(k−1)). The system thus takes a maximum of $\frac{2^{P} - 1}{3}$

cycles to test all non-zero elements of GF(2⁶).

D. A Chien Search System With Fewer Simplified Multipliers

To further optimize the system of FIG. 6, various properties of subfields are utilized.

The updated coefficients are, by definition, elements of GF(2³)², or more generally, GF(2^(mn)). Accordingly, the updated coefficient σ_(j) ^((k)) can be expressed as an n-term expression of elements of GF(2^(m)), where (k) is a time index. In the example, σ_(j) ^((k)) can be expressed as β^(i)x+β^(q), where β^(i) and β^(q) are elements of GF(2³). The updated coefficient in matrix form is $\sigma_{j}^{(k)} = {\begin{bmatrix} \beta^{i} \\ \beta^{q} \end{bmatrix} = \begin{bmatrix} \sigma_{j,1}^{(k)} \\ \sigma_{j,0}^{(k)} \end{bmatrix}}$

When σ_(j) ^((k)) is multiplied by an element γ^(r*n), where $r = \frac{2^{n\quad m} - 1}{2^{m} - 1}$

and γ^(r*n) is of the form (0,β^(W)), the result is ${\sigma_{j}^{(k)}*\gamma^{r*n}} = {\begin{bmatrix} {\sigma_{j,1}^{(k)}*\gamma^{r*n}} \\ {\sigma_{j,0}^{(k)}*\gamma^{r*n}} \end{bmatrix} = \begin{bmatrix} {\sigma_{j,1}^{(k)}*\beta^{w}} \\ {\sigma_{j,0}^{(k)}*\beta^{w}} \end{bmatrix}}$

In the example, ${r = {\frac{2^{6} - 1}{2^{3} - 1} = 9}},\quad {\gamma^{r*n} = {\gamma^{9*2} = {\gamma^{18} = {\left( {000,011} \right) = \beta^{3}}}}},$

and the complete multiplication operation involves individual multiplications of σ_(j,1) and σ_(j,2), which are elements of GF(2^(m)), by β³, which is also an element of GF(2^(m)).

The testing of elements of the form γ^(r*n) as roots can be performed by evaluating n=2 expressions using multiplication operations over GF(2³) as follows:

σ₁ ^((k))(x)=σ_(t,1) x ^(t)+σ_(t−1,1) x ^(t−1)+ . . . +σ_(1,1) x+σ _(0,1)

σ₀ ^((k))(x)=σ_(t,0) x ^(t)+σ_(t−1,0) x ^(t−1)+ . . . +σ_(1,0) x+σ _(0,0)

where the time index has been dropped for ease of understanding.

Assuming the degree of the error locator polynomial is greater than v=2^(m)−1, that is, greater than the number of non-zero elements of GF(2^(m)), the expression σ₀(x) can be rewritten:

(σ_(0,0)+σ_(v,0)+σ_(2v,0)+ . . . )x ⁰+(σ_(1,0)+σ_(v+1,0)+σ_(2v+1)+ . . . )x ¹+ . . . +(σ_(v−1,0) σ_(2v−1,0)+ . . . )x ^(v−1)

since x^(v)=1 in GF(2^(m)). A single element or γ^(r*n)*γ^(i) of GF(2⁶) can be tested as a “roof” of the degree t polynomial σ₀(x) using v−1=2^(m)−2 GF(2^(m)) constant multipliers over GF(2^(m)). The same number of multipliers are needed to test the same element as a “root” of σ₁(x). Accordingly, a total of 2(v−2) constant multipliers over GF(2^(m)) is needed to test an element of GF(2⁶) as a root of σ(x). The number of multipliers thus depends on the selected subfield, and not on the degree of σ(x). If the subfield GF(2^(n)) is selected instead of GF(2^(m)), the system requires m(2^(n)−2) multipliers that operate over GF(2^(n)).

In the example, the roots of an error locator polynomial σ(x)=σ₁₃x¹³+σ₁₂x¹²+σ₁₁x¹¹+σ₁₀x¹⁰+σ₉x⁹+σ₈x⁸+σ₇x⁷+σ₆x⁶+σ₅x⁵+σ₄x⁴+σ₃x³+σ₂x²+σ₁x¹+σ₀ can be determined using the expressions σ₁(x) and σ₀(x) and multiplication operations over GF(2³). In GF(2³), x⁷=1, and the expression σ₀(x) can be rewritten as

σ₀(x)=σ_(13,0) x ⁶+σ_(12,0) x ⁵+σ_(11,0) x ⁴+σ_(10,0) x ³+σ_(9,0) x ²+σ_(8,0) x ¹+σ_(7,0) x ⁰+σ_(6,0) x ⁶+σ_(5,0) x ⁵+σ_(4,0) x ⁴+σ_(3,0) x ³+σ_(2,0) x ²+σ_(1,0) x+σ _(0,0)

Combining like terms, the expression becomes:

σ₀(x)=(σ_(13,0)+σ_(6,0))x ⁶+(σ_(12,0)+σ_(5,0))x ⁵+(σ_(11,0)+σ_(4,0))x ⁴+(σ_(10,0)+σ_(3,0))x ³+(σ_(9,0)+σ_(2,0))x ²+(σ_(8,0)+σ_(7,0))x+σ _(0,0)

A system for iteratively testing an element of the form (γ^(r*n))*γ^(i) as a root of σ(x) is then implemented using v−1 GF(2³) multipliers for each of σ₁(x) and σ₀(x), or 12 GF(2³) multipliers. The GF(2³) multipliers are less complex than GF(2⁶) multipliers, requiring fewer XOR gates, and the system is thus less complex than the system of FIG. 1, which uses t−1 or 12 GF(2⁶) multipliers.

FIG. 7 depicts a Chien search system for use over GF(2³)², where roots of the form (γ^(r*n))*γ^(i) are tested. The coefficient terms σ′_(0,0); σ′_(1,0)=(σ_(8,0)+σ_(7,0)) . . . σ′_(6,0)=(σ_(13,0)+σ_(6,0)) and σ′_(0,1)=σ_(0,1); σ′_(1,1)=(σ_(8,1)+σ_(7,1)) . . . σ′_(6,1)=(σ_(13,1)+σ_(6,1)) are supplied through registers 68 to multipliers 70, which multiply the coefficient terms by the appropriate elements of GF(2^(m)). The multipliers in the exemplary system are thus γ¹⁸=β³ for the x term; (γ¹⁸)²=β⁶ for the x² term; (γ¹⁸)³=β² for the x³ term; (γ¹⁸)⁴=β⁵ of the x⁴ term; (γ¹⁸)⁵=β for the x⁵ term and (γ¹⁸)⁶=β⁴ for the x⁶ term. The multipliers for the terms of σ₁(x) are multipliers 70 that correspond to the elements γ^(j) of the form (β¹, 0) such as γ⁶⁰=(β³, 0), and so forth. The summing circuit 72 produces the sum or sums that are examined to test the elements of GF(2⁶) as possible “roots” of σ₀(x) and σ₁(x). The roots of σ(x) are those elements γ^(i)(γ^(r*n))=β^(a)x+β^(b) that produce sums of zero for both σ₁(β^(a)) and σ₀(β^(b)), or a sum σ₁(β^(a))+σ₂(β^(b)) of zero.

To test two elements γ¹*γ^(r*n) and γ^(i)*(γ^(r*n))² simultaneously, the system includes an additional set of 2(v−1) constant GF(2³) multipliers with v−1 multipliers used to test γ^(i)*(γ^(r*n))² as a root of σ₀(x) and v−1 multipliers used to test γ^(i)*(γ^(r*n))² as a root of σ₁(x). If the element γ^(i)*γ^((r*n)3) is also simultaneously tested, an additional set of 2(v−1) constant GF(2³) multipliers are included, and so forth. A system to simultaneously test three possible roots over GF(2³)² is depicted in FIG. 8. The coefficient terms of σ₁(x) and σ₀(x) are supplied to registers 78 and the terms are iteratively multiplied in multipliers 80 by the appropriate elements of GF(2^(m)) to simultaneously test as roots elements γ^(i)*γ^(r*n), γ^(i)*(γ^(r*n))² and γ^(i)*(γ^(r*n))³. The results are added in the appropriate GF summing circuits 82, to produce three sums, one for each element tested. If a sum is zero, the associated element is determined to be a root. The system, which includes 36 GF(2³) multipliers, is less complex than a system that requires 3(t−1) GF(2⁶) multipliers, or in the example, 36 GF(2⁶) multipliers. The system of FIG. 8 can simultaneously test as roots a maximum of 2^(m)−1 elements if GF(2^(m)) is the selected subfield, or 2^(n−)1 elements if GF(2^(n)) is the selected subfield.

The multipliers 80 of FIG. 8 can be simplified by combining the three multipliers associated with a given register 78 into a single multiplier and eliminating various adders, or XOR gates, as discussed above with reference to FIG. 3.

The system of FIG. 8 can be used with other GF(2⁶) that are generated using different polynomials and have elements α^(i) that are not isomorphic to γ^(k). The elements of the form (0, β^(a)) and (β^(b), 0) of the field of interest are mapped to the various elements (γ^(r*n))^(j), and when a root is found using the system, an inverse mapping or transformation is then performed to determine the corresponding element of the GF(2⁶) of interest. The inverse mapping is performed with a constant multiplier similar to the multiplier discussed above with reference to FIG. 2. This system is thus still less complex than a conventional system that uses t−1 GF(2⁶) multipliers to simultaneously test each additional element of GF(2⁶). 

What is claimed is:
 1. A system for determining the locations of erroneous symbols in a data code word over GF(2^(P)) by simultaneously testing j elements of GF(2^(P)) as possible roots of an associated degree-t error locator polynomial σ(x), the system including: A. a plurality of multipliers that simultaneously produce coefficients of the terms of σ(x) for j elements of GF(2^(P)); B. summing circuitry that adds together the respective terms of σ(x) for each of j elements tested and produces j corresponding sums; C. evaluation circuitry that for each element tested evaluates the associated sum, and for each element that is a root associates with the element an erroneous data code word symbol; and D. a feedback system that at a next clock cycle supplies the coefficients of σ(x) for one element tested in the previous clock cycle to the multipliers, such that the multipliers simultaneously produce updated coefficients associated with j additional elements of GF(2^(P)).
 2. The system of claim 1 wherein the system tests in a clock cycle k=1 the elements x=α², (α²)², (α²)³ . . . (α²)^(j); and in clock cycles k=2 . . . (2^(P)−1)/j tests the elements α²*(α²)^(j(k−1)), (α²)²*(α²)^(j(k−1)) . . . (α²)^(i)*(α²)^(j(k−1)).
 3. The system of claim 2 wherein the system includes t−1 multipliers over GF(2^(P)) and each multiplier includes one or more adders that are configured in accordance with precomputed terms of σ(x) for x=α², (α²)², (α²)³ . . . (α²)^(j) based on weight-one elements of GF(2^(P)).
 4. The system of claim 3 wherein duplicated adders, or XOR gates, are eliminated by producing updated coefficients as combinations of other updated coefficients.
 5. The system of claim 1 wherein the system includes t−1 multipliers that multiply elements of GF(2^(P)) by elements of GF(2^(m)) to produce the coefficients of σ(x).
 6. The system of claim 1 wherein the system includes i. n(2^(m)−2) multipliers over GF(2^(m)), where P=nm, each multiplier producing a coefficient of a term of one of n expressions of σ(x); and ii. the summing circuitry produces one or more sums that correspond to the n expressions for σ(x).
 7. The system of claim 6 further including iii. multiple sets of n(2^(m)−2) multipliers to simultaneously test multiple elements of GF(2^(mn)) as possible roots, and iv. the summing circuitry produces for each element tested a corresponding one or more sums.
 8. A system for determining the locations of erroneous symbols in a data code word over GF(2^(mn)) by testing elements of GF(2^(mn)) as possible roots of an associated degree-t error locator polynomial σ(x), the system including: A. n(2^(m)−2) multipliers over GF(2^(m)), each multiplier producing a coefficient of a term of one of n expressions of σ(x) for x=α^(i); B. summing circuitry that produces one or more sums that correspond to the n expressions for σ(x); C. evaluation circuitry that determines that an element is a root if the associated one or more sums are equal to zero, the evaluation circuitry associating the root with an erroneous data code word symbol; and D. a feedback system that at a next clock cycle supplies to the respective multipliers coefficients produced in the previous clock cycle, wherein the system tests a next element as a possible root.
 9. The system of claim 8 further including E. multiple sets of n(2^(m)−2) multipliers to simultaneously test multiple elements of GF(2^(mn)) as possible roots, and F. the summing circuitry produces for each element a corresponding sum. 